can anyone help with this, any direction could be helpfull?
I've tried using the fact that $ G_1 $ satisfies that it's planar and is triangle free because G is. So we should have $|E_1| \leq 2|V|-4 $ ...

I was reading the slides of a talk by Tom Leinster.
I have trouble understanding the last line of page 17 (pages 1-15 are irrelevant and can be skipped). Could someone please explain it to me?
If I ...

the question:
what is the sum of heights of the vertices of a perfect binary tree (n vertices, the height of a leaf is 0)?
explain shortly.
a. $\theta(logn) $ b. $\theta(nlogn) $ c. $\theta(n) $ d. ...

I'm working on this project for an oil and gas company. One of the main features is a visualization of their pipe network.
I'm trying to create a tree of all possible paths. The only limit i have to ...

Can someone give a direct proof (NOT an inductive proof) showing that a subtree rooted at any node $x$ in a red black tree has at least $2^{bh(x)}-1$ internal nodes ?
$bh(x)$ means the black height ...

In all my books and articles about "graph theory", I didn't find the definition of "directed spanning tree". Could you please give this definition and the reference?
How to judge if a directed graph ...

Suppose I have the labeled trees $T_{1}, \ldots, T_{n}$ with $b_{1}, \ldots, b_{n}$ vertices respectively. I would like to know how many ways I can compose a tree from these trees by using all trees? ...

In graph theory, given a rooted tree $T$ and a node $a \in V(T)$, is there a standard way to refer to the set of all children of $a$? I have seen $CHILDREN_T(a)$ being used, but this seem quite clumsy ...

Let $T$ be a labeled tree on the set of vertices $\{1,...,n\}$, and its sequence of degrees is $d_1,...,d_n$. Prove that for all $1\le i \le n$ the number of appearances in $F(T)$ (Prufer sequence) ...

I can store any undirected simple graph N vertices using $b = (N-1)N/2$ bits, by creating a mask of the edges on the upper diagonal of the adjacency matrix. For example the adjacency matrix of $K_3$ ...

A question regarding rooted plane trees bothers me. We know that the number of rooted plane trees with $n$ nodes equals to $n-{th}$ Catalan number, that is $|Tn| = Cn$.
But what is this number if we ...

Given any undirected connected graph. If we redefine the weight of a spanning tree to the maximum weight of an edge (if the largest weight is 10 the weight of the tree is 10) are there any cases where ...

Let $K_{a,b}$ be the complete bipartite graph. Show that
$K_{a,b}$ is a tree if and only if $a = 1$ or $b = 1$.
The way my professor showed us for a complete graph is as below. I just don't know how ...

Suppose I have n nodes, how can I find the max and min height of a tree?
I've seen varying statements for the min height such as log2 (n) and log2 (n+1) but I wasn't sure which was correct and I am ...